Conic Sections: A Rejuvenated Exploration

Conic sections are defined as nondegenerate curves that are generated by the intersections of a plane with one or two nappes of a cone (Weisstein, n.d.). A cone is generated by a moving straight line that continuously touches a fixed curve and goes through a fixed point that is not in the plane of the curve. We call this moving line the generatrix, the fixed curve the directrix, and the fixed point the vertex (Wentworth, 1913). There are four different types of conic sections: the parabola, the hyperbola, the ellipse, and the circle. The parabola is the curve where the locus of a point moves in a plane such that the distance from a fixed point in the plane is continuously equal to its distance from a fixed line in the plane. The hyperbola is the curve in which the locus of a point moves in a plane such that the difference of its distances from the two fixed points in the plane is constant. The ellipse is the curve where the locus of a point moves in a plane such that the sum of those distances from the two fixed points are constant (Wentworth, 1913). The fourth conic section, the circle, can be viewed as a special case of the ellipse. These fixed points are called foci and the lines that link a point of the curve to the foci is the foci radii. The circle exists when the distance of each foci radius is equal to each other (Purcell, 1958). In this website we will explore the properties of each of these figures as well as the properties of conic sections in general.